Optimal. Leaf size=40 \[ -\frac {(c-b c x)^{n+1} \, _2F_1(-m,n+1;n+2;1-b x)}{b c (n+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {65} \[ -\frac {(c-b c x)^{n+1} \, _2F_1(-m,n+1;n+2;1-b x)}{b c (n+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rubi steps
\begin {align*} \int (b x)^m (c-b c x)^n \, dx &=-\frac {(c-b c x)^{1+n} \, _2F_1(-m,1+n;2+n;1-b x)}{b c (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 1.10 \[ \frac {x (b x)^m (1-b x)^{-n} (c-b c x)^n \, _2F_1(m+1,-n;m+2;b x)}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-b c x + c\right )}^{n} \left (b x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-b c x + c\right )}^{n} \left (b x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (b x \right )^{m} \left (-b c x +c \right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-b c x + c\right )}^{n} \left (b x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (b\,x\right )}^m\,{\left (c-b\,c\,x\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.50, size = 37, normalized size = 0.92 \[ \frac {b^{m} c^{n} x x^{m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {b x e^{2 i \pi }} \right )}}{\Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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